doc
... Explanation: Clearly xFx is false in this case, since there are numbers that are not prime, for example 6. So, then xFx→xGx is true. Now, note that x(Fx→Gx) is false since there is a number n such that ~(Fn→Gn), for example 5. So, the above interpretation shows that the sequent is invalid. ...
... Explanation: Clearly xFx is false in this case, since there are numbers that are not prime, for example 6. So, then xFx→xGx is true. Now, note that x(Fx→Gx) is false since there is a number n such that ~(Fn→Gn), for example 5. So, the above interpretation shows that the sequent is invalid. ...
Predicate logic
... Definition: integer a is odd iff a = 2m + 1 for some integer m Let a, b ∈ Z s.t. a and b are odd. Then by definition of odd a = 2m + 1.m ∈ Z and b = 2n + 1.n ∈ Z So ab = (2m + 1)(2n + 1) = 4mn + 2m + 2n + 1 = 2(2mn + m + n) + 1 and since m, n ∈ Z it holds that (2mn + m + n) ∈ Z, so ab = 2k + 1 for s ...
... Definition: integer a is odd iff a = 2m + 1 for some integer m Let a, b ∈ Z s.t. a and b are odd. Then by definition of odd a = 2m + 1.m ∈ Z and b = 2n + 1.n ∈ Z So ab = (2m + 1)(2n + 1) = 4mn + 2m + 2n + 1 = 2(2mn + m + n) + 1 and since m, n ∈ Z it holds that (2mn + m + n) ∈ Z, so ab = 2k + 1 for s ...
Tactics for Separation Logic Abstract Andrew W. Appel INRIA Rocquencourt & Princeton University
... proving an imperative program, much of the reasoning is not about memory cells but concerns the abstract mathematical objects that the program’s data structures represent. Lemmas about those objects are most conveniently proved in a general-purpose higher-order logic, especially when there are large ...
... proving an imperative program, much of the reasoning is not about memory cells but concerns the abstract mathematical objects that the program’s data structures represent. Lemmas about those objects are most conveniently proved in a general-purpose higher-order logic, especially when there are large ...
S. P. Odintsov “REDUCTIO AD ABSURDUM” AND LUKASIEWICZ`S
... to finish an investigation of the class of Lj-extensions with an attempt to overcome it. We try to do it by emerging the class of Lj-extensions in a more general class of paraconsistent logics and pointing out some special property distinguishing extensions of minimal logic in the latter class. We su ...
... to finish an investigation of the class of Lj-extensions with an attempt to overcome it. We try to do it by emerging the class of Lj-extensions in a more general class of paraconsistent logics and pointing out some special property distinguishing extensions of minimal logic in the latter class. We su ...
preprint - Open Science Framework
... Subject, and therefore true, is that the Creating Subject at some point constructed objects and relations between them that made that proposition evident. This is why Brouwer wrote, ‘truth is only in reality i.e. in the present and past experiences of conciousness … there are no non-experienced trut ...
... Subject, and therefore true, is that the Creating Subject at some point constructed objects and relations between them that made that proposition evident. This is why Brouwer wrote, ‘truth is only in reality i.e. in the present and past experiences of conciousness … there are no non-experienced trut ...
1Propositional Logic - Princeton University Press
... The first representation is technically correct (ignoring the “use” instruction), but useless. The idea is to formalize a sentence in as finegrained an encoding as is possible with the logic at hand. The second and third representations do this. Notice that there are three different English expressi ...
... The first representation is technically correct (ignoring the “use” instruction), but useless. The idea is to formalize a sentence in as finegrained an encoding as is possible with the logic at hand. The second and third representations do this. Notice that there are three different English expressi ...
The Pure Calculus of Entailment Author(s): Alan Ross Anderson and
... i-th step Ai in the proofs above there is associated first a number of vertical lines to the left of Ai (which we shall call the rank of As), secondly a class of formulas (including Ai) which are candidates for application of the rule of repetition to yield a next step for the proof, thirdly a class ...
... i-th step Ai in the proofs above there is associated first a number of vertical lines to the left of Ai (which we shall call the rank of As), secondly a class of formulas (including Ai) which are candidates for application of the rule of repetition to yield a next step for the proof, thirdly a class ...
Truth-Functional Logic
... Parentheses and the Method of Induction A statement like A ∧ B ∨ C is ambiguous: is this the conjunction of A and B ∨ C, or is this a disjunction of A ∧ B and C? We will require that our statements are not ambiguous, and we will use parentheses to do so. So, if this statement was meant to be a conj ...
... Parentheses and the Method of Induction A statement like A ∧ B ∨ C is ambiguous: is this the conjunction of A and B ∨ C, or is this a disjunction of A ∧ B and C? We will require that our statements are not ambiguous, and we will use parentheses to do so. So, if this statement was meant to be a conj ...
Notes on Modal Logic - Stanford University
... • Alethic Reading: 2ϕ means ‘ϕ is necessary’ and 3ϕ means ‘ϕ is possible’. • Deontic Reading: 2ϕ means ‘ϕ is obligatory’ and 3ϕ means ‘ϕ is permitted’. In this literature, typically ‘O’ is used instead of ‘2’ and ‘P ’ instead of ‘3’. • Epistemic Reading: 2ϕ means ‘ϕ is known’ and 3ϕ means ‘ϕ is cons ...
... • Alethic Reading: 2ϕ means ‘ϕ is necessary’ and 3ϕ means ‘ϕ is possible’. • Deontic Reading: 2ϕ means ‘ϕ is obligatory’ and 3ϕ means ‘ϕ is permitted’. In this literature, typically ‘O’ is used instead of ‘2’ and ‘P ’ instead of ‘3’. • Epistemic Reading: 2ϕ means ‘ϕ is known’ and 3ϕ means ‘ϕ is cons ...
page 135 ADAPTIVE LOGICS FOR QUESTION EVOCATION
... to establish, for some question Q and some set of sentences Γ, that none of the direct answers to Q is derivable from Γ, and hence, to establish that Q is evoked by Γ. There are several ways to deal with this lack of a positive test. One way is to consider only decidable theories. For such theories, ...
... to establish, for some question Q and some set of sentences Γ, that none of the direct answers to Q is derivable from Γ, and hence, to establish that Q is evoked by Γ. There are several ways to deal with this lack of a positive test. One way is to consider only decidable theories. For such theories, ...
Soundness and Completeness for Sentence Logic Trees
... between a semantic concept-validity-and a corresponding syntactic concept-proofs. Let's be explicit about what counts as a proof in the tree method: Given some premises and a conclusion, a tree method proof is a closed tree (a tree with all its branches closed) which has the premises and negation of ...
... between a semantic concept-validity-and a corresponding syntactic concept-proofs. Let's be explicit about what counts as a proof in the tree method: Given some premises and a conclusion, a tree method proof is a closed tree (a tree with all its branches closed) which has the premises and negation of ...
this PDF file
... circular if its conclusion is contained in its premises, or if it assumes — either explicitly or not — what it is trying to prove (see, for instance, [4]). An example of a circular argument can be found in the context of first year Calculus when one proves that lim ...
... circular if its conclusion is contained in its premises, or if it assumes — either explicitly or not — what it is trying to prove (see, for instance, [4]). An example of a circular argument can be found in the context of first year Calculus when one proves that lim ...
Classical first-order predicate logic This is a powerful extension of
... 7.2 Truth in a structure (a rough guide) When is a formula true in a structure? • Sun(Heron) is true in M , because HeronM is an object fthat M says is a Sun. We write this as M |= Sun(Heron). Can read as ‘M says Sun(Heron)’. Warning: This is a quite different use of |= from definition 3.1. ‘|=’ is ...
... 7.2 Truth in a structure (a rough guide) When is a formula true in a structure? • Sun(Heron) is true in M , because HeronM is an object fthat M says is a Sun. We write this as M |= Sun(Heron). Can read as ‘M says Sun(Heron)’. Warning: This is a quite different use of |= from definition 3.1. ‘|=’ is ...
A Proof of Nominalism. An Exercise in Successful
... Symbolic logic is a marvelous thing. It allows for an explicit expression of existence, viz. by means of the existential quantifier, and by it only. This is the true gist in Quine’s slogan “to be is to be a value of a bound variable.” Accordingly, one can also formulate explicitly the thesis of nomi ...
... Symbolic logic is a marvelous thing. It allows for an explicit expression of existence, viz. by means of the existential quantifier, and by it only. This is the true gist in Quine’s slogan “to be is to be a value of a bound variable.” Accordingly, one can also formulate explicitly the thesis of nomi ...
Chapter 5.5
... 3) The measure of any obtuse angle is greater than 90. The sum of any two obtuse angles is greater than 180. This is a CONTRADICTION to the theorem that states that the sum of the angles of any triangle = 180. 4) Therefore, my assumption is false. A triangle cannot have two obtuse angles. ...
... 3) The measure of any obtuse angle is greater than 90. The sum of any two obtuse angles is greater than 180. This is a CONTRADICTION to the theorem that states that the sum of the angles of any triangle = 180. 4) Therefore, my assumption is false. A triangle cannot have two obtuse angles. ...
Lecture notes for Chapter 1
... Its truth value depends on the value of y, but this value is not specified. We call this type of statement a propositional function or open sentence. Spring 2003 ...
... Its truth value depends on the value of y, but this value is not specified. We call this type of statement a propositional function or open sentence. Spring 2003 ...
Formal deduction in propositional logic
... ’Contrariwise,’ continued Tweedledee, ’if it was so, it might be; and if it were so, it would be; but as it isn’t, it ain’t. That’s logic.’ (Lewis Caroll, “Alice in Wonderland”) Formal deduction in propositional logic ...
... ’Contrariwise,’ continued Tweedledee, ’if it was so, it might be; and if it were so, it would be; but as it isn’t, it ain’t. That’s logic.’ (Lewis Caroll, “Alice in Wonderland”) Formal deduction in propositional logic ...
Classical first-order predicate logic This is a powerful extension
... 7.2 Truth in a structure (a rough guide) When is a formula true in a structure? • Sun(Heron) is true in M , because HeronM is an object fthat M says is a Sun. We write this as M |= Sun(Heron). Can read as ‘M says Sun(Heron)’. Warning: This is a quite different use of |= from definition 3.1. ‘|=’ is ...
... 7.2 Truth in a structure (a rough guide) When is a formula true in a structure? • Sun(Heron) is true in M , because HeronM is an object fthat M says is a Sun. We write this as M |= Sun(Heron). Can read as ‘M says Sun(Heron)’. Warning: This is a quite different use of |= from definition 3.1. ‘|=’ is ...
... is valid iff all substitution instances of the formula are valid. However, when it comes to inference, this distinction is crucial. For example, if we did not consider substitution instances, then p F,, false would hold, where p is a primitive proposition, since p is not valid.4 However, when viewed ...
Logic Programming, Functional Programming, and Inductive
... Sound (or partially correct) means that each successful goal in an execution is permitted by the specification. Complete means that each permitted goal will succeed during execution. Specifications are still written in some sort of logical formalism even when we regard programs as inductive definiti ...
... Sound (or partially correct) means that each successful goal in an execution is permitted by the specification. Complete means that each permitted goal will succeed during execution. Specifications are still written in some sort of logical formalism even when we regard programs as inductive definiti ...
Logic for Gottlob Frege and Bertrand Russell:
... Propositions are the semantic content of thought, and propositions stand essentially in inferential relations to each other. “The” logico-philosophical question: how are inferential relations essential to thought? I. Frege: formal logic can answer this question by developing a logical notation (Begr ...
... Propositions are the semantic content of thought, and propositions stand essentially in inferential relations to each other. “The” logico-philosophical question: how are inferential relations essential to thought? I. Frege: formal logic can answer this question by developing a logical notation (Begr ...
GLukG logic and its application for non-monotonic reasoning
... the replacement and deduction theorems. It should be expressive enough. If possible it should be finitely axiomatizable and somehow close to some constructive logic. If possible the logic should be many-valued. 2. Definition 1 should give the same result to our basic program B as the stable semantic ...
... the replacement and deduction theorems. It should be expressive enough. If possible it should be finitely axiomatizable and somehow close to some constructive logic. If possible the logic should be many-valued. 2. Definition 1 should give the same result to our basic program B as the stable semantic ...
SITUATIONS, TRUTH AND KNOWABILITY — A
... is quite implausible that it should be possible to establish Fitch's strong conclusion: For each agent who is not omniscient, there is a true proposition which the agent cannot know. ...
... is quite implausible that it should be possible to establish Fitch's strong conclusion: For each agent who is not omniscient, there is a true proposition which the agent cannot know. ...